The propagation of transient elastic waves in a three-dimensional multilayered medium subjected to general dynamic loadings is analyzed in this study. One of the objectives of this study is to develop an effective analytical method for determining transient full-field solutions in the multilayered medium subjected to nonaxisymmetric dynamic loadings. The wave field is decomposed into two parts; one is the P-SV system of waves and the other is the SH system, each of which propagates independently in a horizontally layered medium. A simple relationship between solutions of two- and three-dimensional problems in the Fourier transform domain is also presented. A matrix solution in the triple transform domain is obtained by Brownwich expansion and the inverse transform is inverted by Cagniard’s method. The matrix representation of the solution can be easily used to calculate the transient response of the multilayered medium without tracing the ray path manually. The formulas presented in this study enable us to calculate the long-time transient responses of the multilayered medium for three-dimensional configuration. The theoretical transient solution for a layer overlaying a half-space subjected to an arbitrarily oriented point force is presented in detail. The numerical calculations are compared with experimental measurements for the surface displacement response on a plate and a layer overlaying a half-space subjected to an axisymmetric point force. The numerical investigation is also carried out for nonaxisymmetric loading problem of a layer overlaying a half-space.

1.
Lamb
,
H.
, 1904, “
On the Propagation of Tremors Over the Surface of an Elastic Solid
,”
Philos. Trans. R. Soc. London
0962-8428,
A203
, pp.
1
42
.
2.
Ewing
,
W. M.
,
Jardetzky
,
W. S.
, and
Press
,
F.
, 1957,
Elastic Waves in Layered Media
,
McGraw-Hill
, New York.
3.
Brekhovskikh
,
L. M.
, 1980,
Waves in Layered Media
,
Academic
, New York.
4.
Thomson
,
W. T.
, 1950, “
Transmission of Elastic Waves through a Stratified Solid Medium
,”
J. Appl. Phys.
0021-8979,
21
, pp.
89
93
.
5.
Haskell
,
N.
, 1953, “
The Dispersion of Surface Waves on Multi-layered Media
,”
Bull. Seismol. Soc. Am.
0037-1106,
43
, pp.
17
34
.
6.
Gilbert
,
F.
, and
Backus
,
G. E.
, 1966, “
Propagator Matrices in Elastic Wave and Vibration Problems
,”
Geophys. J.
0275-9128,
31
, pp.
326
332
.
7.
Kennett
,
B. L. N.
, and
Kerry
,
N. J.
, 1979, “
Seismic Waves in a Stratified Half Space
,”
Geophys. J. R. Astron. Soc.
0016-8009,
44
, pp.
557
583
.
8.
Kennett
,
B. L. N.
, 1983,
Seismic Wave Propagation in Stratified Media
,
Cambridge University Press
, Cambridge, UK.
9.
Ma
,
C. C.
, and
Huang
,
K. C.
, 1996, “
Analytical Transient Analysis of Layered Composite Medium Subjected to Dynamic Inplane Impact Loadings
,”
Int. J. Theor. Phys.
0020-7748,
33
, pp.
4223
4238
.
10.
Miklowitz
,
J.
, 1962, “
Transient Compressional Waves in an Infinite Elastic Plate or Elastic Layer Overlaying a Rigid Half-space
,”
ASME J. Appl. Mech.
0021-8936,
29
, pp.
53
60
.
11.
Kundu
,
T.
, and
Mal
,
A. K.
, 1985, “
Elastic Waves in a Multilayered Solid due to a Dislocation Source
,”
Wave Motion
0165-2125,
7
, pp.
459
471
.
12.
Mal
,
A. K.
, and
Lih
,
S. S.
, 1992, “
Elastodynamic Response of a Unidirectional Composite Laminate to Concentrated Surface Loads: Part I
,”
ASME J. Appl. Mech.
0021-8936,
59
, pp.
878
886
.
13.
Lih
,
S. S.
, and
Mal
,
A. K.
, 1992, “
Elastodynamic Response of a Unidirectional Composite Laminate to Concentrated Surface Loads: Part II
,”
ASME J. Appl. Mech.
0021-8936,
59
, pp.
887
892
.
14.
Cetinkaya
,
C.
, and
Vakaris
,
A. F.
, 1996, “
Transient Axisymmetric Stress Wave Propagation in Weakly Coupled Layered Structures
,”
J. Sound Vib.
0022-460X,
194
, pp.
389
416
.
15.
Banerjee
,
S.
,
Prosser
,
W. H.
, and
Mal
,
A. K.
, 2005, “
Calculation of the Response of a Composite Plate to Localized Dynamic Surface Loads Using a New Wave Number Integral Method
,”
ASME J. Appl. Mech.
0021-8936,
72
, pp.
18
24
.
16.
Prosser
,
W. H.
,
Hamstad
,
M. A.
,
Gary
,
J.
, and
O’Gallagher
,
A.
, 1999, “
Finite Element and Plate Theory Modeling of Acoustic Emission Waveforms
,”
J. Nondestruct. Eval.
0195-9298,
18
, pp.
83
90
.
17.
Cagniard
,
L.
, 1939,
Reflexion et Refraction des ondes Seismiques Progressives
,
Cauthiers-Villars
, Paris;
Translated into English and revised by
E. A.
Flinn
and
C. H.
Dix
, 1962
Reflection and Refraction of Progressive Seismic Waves
,
McGraw-Hill
, New York.
18.
Pao
,
Y. H.
, and
Gajewski
,
R.
, 1977, “
The Generalized Ray Theory and Transient Responses of Layered Elastic Solids
,”
Physical Acoustics
,
Mason
,
W. P.
, and
Thurston
,
R. N.
, eds.,
Academic
, New York, Vol.
13
, Chap. 6, pp.
184
266
.
19.
Mencher
,
A. G.
, 1953, “
Epicentral Displacement Caused by Elastic Waves in an Infinite Slab
,”
J. Appl. Phys.
0021-8979,
24
, pp.
1240
1246
.
20.
Knopoff
,
L.
, 1958, “
Surface Motion of a Thick Plate
,”
J. Appl. Phys.
0021-8979,
29
, pp.
661
670
.
21.
Davids
,
N.
, 1959, “
Transient Analysis of Stress-Wave Penetration in Plate
,”
ASME J. Appl. Mech.
0021-8936,
26
, pp.
651
660
.
22.
Weaver
,
R. L.
, and
Pao
,
Y. H.
, 1982, “
Axisymmetric Elastic Waves Excited by a Point Source in a Plate
,”
ASME J. Appl. Mech.
0021-8936,
49
, pp.
821
836
.
23.
Davids
,
N.
, and
Lawhead
,
W.
, 1965, “
Transient Analysis of Oblique Impact of Plates
,”
J. Mech. Phys. Solids
0022-5096,
13
, pp.
199
212
.
24.
Ceranoglu
,
A. N.
, and
Pao
,
Yih-Hsing
, 1981, “
Propagation of Elastic Pulses and Acoustic Emission in a Plate
,”
ASME J. Appl. Mech.
0021-8936,
48
, pp.
125
147
.
25.
Achenbach
,
J. D.
, and
Xu
,
Y.
, 1999, “
Use of Elastodynamic Reciprocity to Analyze Point-Load Generated Axisymmetric Waves in a Plate
,”
Wave Motion
0165-2125,
30
, pp.
57
67
.
26.
Spencer
,
T. W.
, 1960, “
The Method of Generalized Reflection and Transmission Coefficients
,”
Geophys. J.
0275-9128,
25
, pp.
625
641
.
27.
Pekeris
,
C. L.
,
Alterman
,
Z.
,
Abramovici
,
F.
, and
Jarosh
,
H.
, 1965, “
Propagation of a Compressional Pulse in a Layered Solid
,”
Rev. Geophys.
0096-1043,
3
, pp.
25
47
.
28.
Aki
,
K.
, and
Richard
,
P. G.
, 1980,
Quantitative Seismology: Theory and Methods
,
Freeman
, San Fransico.
29.
Van Der Hijden
,
J. H. M. T.
, 1987,
Propagation of Transient Elastic Waves in Stratified Anisotropic Media
,
Springer Verlag
, New York.
30.
Müller
,
G.
, 1968, “
Theoretical Seismograms for Some Types of Point-Source in Layered Media, Part I: Theory
,”
Z. Geophys.
0044-2801,
34
, pp.
15
35
.
31.
Müller
,
G.
, 1968, “
Theoretical Seismograms for Some Types of Point-Source in Layered Media, Part II: Numerical Calculations
,”
Z. Geophys.
0044-2801,
34
, pp.
147
162
.
32.
Müller
,
G.
, 1969, “
Theoretical Seismograms for Some Types of Point-Source in Layered Media, Part III: Single Forces and Dipole Sources of Arbitrary Orientation
,”
Z. Geophys.
0044-2801,
35
, pp.
347
371
.
33.
Lee
,
G. S.
, and
Ma
,
C. C.
, 2000, “
Transient Elastic Waves Propagating in a Multi-Layered Medium Subjected to In-Plane Dynamic Loadings. I. Theory
,”
Proc. R. Soc. London, Ser. A
1364-5021,
456
, pp.
1355
1374
.
34.
Ma
,
C. C.
, and
Lee
,
G. S.
, 2000, “
Transient Elastic Waves Propagating in a Multi-Layered Medium Subjected to In-Plane Dynamic Loadings. II. Numerical Calculation and Experimental Measurement
,”
Proc. R. Soc. London, Ser. A
1364-5021,
456
, pp.
1375
1396
.
35.
Su
,
X. Y.
,
Tian
,
T. Y.
, and
Pao
,
Y. H.
, 2002, “
Application of the Reverberation-Ray Matrix to the Propagation of Elastic Waves in a Layered Solid
,”
Int. J. Solids Struct.
0020-7683,
39
, pp.
5447
5463
.
36.
Achenbach
,
J. D.
, 1973,
Wave Propagation in Elastic Solid
,
North-Holland
, Amsterdam.
37.
Ma
,
C. C.
, and
Lee
,
G. S.
, 1999, “
Theoretical Analysis, Numerical Calculation and Experimental Measurement of Transient Elastic Waves in Strips Subjected to Dynamic Loadings
,”
Int. J. Solids Struct.
0020-7683,
36
, pp.
3541
3564
.
38.
Sachse
,
W.
, and
Hsu
,
N. N.
, 1979, “
Ultrasonic Transducers for Materials Testing and Their Characterization
,”
Physical Acoustics
,
Manson
,
W. P.
and
Thurston
,
R. N.
, eds.,
Academic
, New York, Vol.
XIV
, Ch. 4, pp.
277
406
.
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